IDOS Energy, Coherence, and Supporting Dynamics — Part 57

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squares and the support squares; $\|T_0u\|_2^2$ is their integral. Frame norms count independent entries once. Writing the velocity coupling as $\langle u_t,Bu\rangle$, its nonzero blocks are $B_{\upsilon,\xi}=2I/\epsilon_c$, $B_{\vartheta,\zeta_W}=c$, $B_{\upsilon,\nu}=\sqrt{\epsilon_c/\kappa_U}H$ and $B_{\zeta,\nu}=\sqrt{C_s/\kappa_U}D_s/W$; $G=B-B^T$. Unbounded multipliers, including $\kappa_G/\mu$, remain combined residuals. The exact domains and equation are

$$\mathcal V=\{u\in L^2:T_0u\in L^2\text{ distributionally}\},\quad \|u\|_{\mathcal V}^2=\|u\|_2^2+\|T_0u\|_2^2,\quad u_{tt}+G u_t+T_0^*T_0u=0,\quad G^T=-G.$$

The entropy gyro entry is $c(x)=\sqrt{b/C_s}\,Wq_0'/q_0\sim\sqrt{b/C_s}|x|^2/(4\varepsilon_pL^2)$. It is locally bounded but globally unbounded. Compact cutoffs commute with multiplication terms and have residual commutator $O(1/R)$; local derivative isolation then proves that compact smooth fields are a form core. Compact smooth linear solutions extend to every finite time collar with speed at most one. Their exact conserved energy gives

$$e(t)^2=\|u_t(t)\|_2^2+\|T_0u(t)\|_2^2=e(0)^2,\qquad \|u(t)\|_2\le\|u(0)\|_2+|t|e(0).$$

Completion yields a unique strongly continuous conservative group on $\mathcal V\times L^2$, bounded by $1+|t|$, with $u\in C(\mathbb R;\mathcal V)\cap C^1(\mathbb R;L^2)$ and $u_{tt}\in C H^{-1}_{\rm loc}$. The generator is precisely the graph closure of $(u,p)\mapsto(p,-T_0^*T_0u-Gp)$ on compact smooth pairs. Here $A$ is the nonnegative Friedrichs operator of the static form. Neither $D(A)\times\mathcal V$ nor a formal maximal generator domain is asserted. The displacement norm defines topology, not an added restoring energy.

Density admissibility is narrower: for $r=\delta q/q_0$, $\sqrt{q_0}r=(c\zeta_W+p_\theta)/\sqrt b$, where $p_\theta$ is the velocity of the normalized phase component. Reconstruction is locally $L^2$ and is globally so for compact smooth solutions, but not on the whole completed space. Disjoint unit-size packets centered at $R_j e_1$, $R_j=2^jL$, with amplitude $a_j=R_j^{-3/2}$ and $\zeta_Z=[(9t-1)/(8\sqrt t(1+t))]\zeta_W x/|x|$ cancel the algebraic support residual. Their squared form norms are bounded by $K(1+R_j^2)a_j^2$, whereas squared density norms exceed $k_0R_j^4a_j^2$. The first sum converges and the second diverges. Stronger density domains need a separate propagation proof. The symmetric-hyperbolic tools used here are those of Métivier, sections 3.3 and 3.6, applied to the displayed action; they supply no physical optical premise.

Smooth coefficient extensions do not give vacuum evolution: positive principal weights degenerate and the unchanged unweighted support penalty contains $-\kappa_GZ^2W^{9/4}$, which has no full-neighborhood $C^3$ extension through vacuum. None of these results supplies global nonlinear stability, a spectral gap, a uniform sharp-layer limit or an infinite-carrier limit.

The Plateau Changes The Support Recoil Constants

For radial $q_0$, write $\langle a\rangle_q=Q^{-1}\int q_0a\,d^{10}x$. At fixed rational positions, carrier jet, rest flow with $U_t=0$ and constant frame with $E_t=0$, the scalar and vector translation Grams are

$$g_W=\frac{Q\omega_{\min}}{212}\left\langle\frac{32t}{5(1+t)^2}\right\rangle_q, \quad g_Z=\frac{Q\omega_{\min}}{212}\left\langle\frac{1+9t^2}{(1+t)^4}\right\rangle_q, \quad g=g_W+g_Z>0.$$

For carrier canonical covector $p$, the unique finite-kinetic-energy support minimum cancelling it has $W_t=-(p/g)\cdot\nabla W$, $Z_t=-(p/g)\cdot\nabla Z$ and cost $|p|^2/(2g)$. This follows by orthogonal

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